• Open Access

Geometry and Response of Lindbladians

Victor V. Albert, Barry Bradlyn, Martin Fraas, and Liang Jiang
Phys. Rev. X 6, 041031 – Published 16 November 2016

Abstract

Markovian reservoir engineering, in which time evolution of a quantum system is governed by a Lindblad master equation, is a powerful technique in studies of quantum phases of matter and quantum information. It can be used to drive a quantum system to a desired (unique) steady state, which can be an exotic phase of matter difficult to stabilize in nature. It can also be used to drive a system to a unitarily evolving subspace, which can be used to store, protect, and process quantum information. In this paper, we derive a formula for the map corresponding to asymptotic (infinite-time) Lindbladian evolution and use it to study several important features of the unique state and subspace cases. We quantify how subspaces retain information about initial states and show how to use Lindbladians to simulate any quantum channels. We show that the quantum information in all subspaces can be successfully manipulated by small Hamiltonian perturbations, jump operator perturbations, or adiabatic deformations. We provide a Lindblad-induced notion of distance between adiabatically connected subspaces. We derive a Kubo formula governing linear response of subspaces to time-dependent Hamiltonian perturbations and determine cases in which this formula reduces to a Hamiltonian-based Kubo formula. As an application, we show that (for gapped systems) the zero-frequency Hall conductivity is unaffected by many types of Markovian dissipation. Finally, we show that the energy scale governing leakage out of the subspaces, resulting from either Hamiltonian or jump-operator perturbations or corrections to adiabatic evolution, is different from the conventional Lindbladian dissipative gap and, in certain cases, is equivalent to the excitation gap of a related Hamiltonian.

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  • Received 31 January 2016

DOI:https://doi.org/10.1103/PhysRevX.6.041031

Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Quantum Information, Science & TechnologyCondensed Matter, Materials & Applied PhysicsStatistical Physics & ThermodynamicsAtomic, Molecular & Optical

Authors & Affiliations

Victor V. Albert1,*, Barry Bradlyn2, Martin Fraas3, and Liang Jiang1

  • 1Departments of Applied Physics and Physics, Yale University, New Haven, Connecticut 06511, USA
  • 2Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08540, USA
  • 3Mathematisches Institut der Universität München, University of Munich, Munich 80333, Germany

  • *valbert4@gmail.com

Popular Summary

Introductory physics courses often focus on systems that are isolated from their environments (e.g., an object falling without air resistance). Besides being necessary for a complete understanding of the systems in question, environmental effects can also steer the systems in favorable directions, such as a parachute using air resistance to prevent an object from falling. In quantum systems, environmental effects can be even more pronounced, and we are on the brink of synthesizing longer-lasting quantum memories, faster quantum computers, and exotic and previously inaccessible quantum phases of matter. However, commonplace environmental effects are known to destroy delicate quantum states. To what extent can the environment be used to control a quantum system without obscuring any of the system’s useful and elegant features? Can some of these features become even richer in the presence of an environment? Here, we comprehensively address these important questions in a theoretical manner.

The simplest model generalizing the notion of a closed quantum system is the Lindblad master equation, or Lindbladian. Lindbladians, unlike Hamiltonians, can evolve all initial states of the system to a desired steady state or steady-state subspace. For this reason, Lindbladians are useful in engineering both exotic phases of matter and protected subspaces for quantum information processing. However, many features of Hamiltonian systems have not yet been generalized or understood to hold true for Lindbladian systems: How do quantum states decay from one subspace into another? How do generic steady-state subspaces respond to being probed by perturbations? What are the geometric properties of these subspaces? We provide Lindbladian generalizations of these and other features. We also quantify how perturbations can cause states to leak out of the steady-state subspaces and show that decay is universal (i.e., one can use decay to simulate any more-general quantum processes). It turns out that common steady-state subspaces do not differ significantly from their Hamiltonian counterparts.

We anticipate that our generalizations of the various concepts from Hamiltonian-based systems to Lindbladians will warrant further theoretical investigation. We also expect that some of our predictions of the properties of open-system subspaces will pave the way for experimental verification.

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Vol. 6, Iss. 4 — October - December 2016

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